View of Sequences for complexes II

SEQUENCES FOR COMPLEXES II

LARS WINTHER CHRISTENSEN

1. Introduction and Notation

This short paper elaborates on an example given in [4] to illustrate an applic- ation of sequences for complexes:

LetRbe a local ring with a dualizing complexD, and letMbe a finitely generatedR-module; then a sequencex¹, . . . , xn is part of a system of parameters forMif and only if it is aRHom_R(M, D)-sequence [4, 5.10].

The final Theorem 3.9 of this paper generalizes the result above in two directions: the dualizing complex is replaced by a Cohen-Macaulay semi- dualizing complex (see [3, Sec. 2] or 3.8 below for definitions), and the finite module is replaced by a complex with finite homology.

Before we can even state, let alone prove, this generalization of [4, 5.10]

we have to introduce and study parameters for complexes. For a finite R- moduleM everyM-sequence is part of a system of parameters for M, so, loosely speaking, regular elements are just special parameters. For a complex X, however, parameters and regular elements are two different things, and kinship between them implies strong relations between two measures of the size ofX: theamplitudeand theCohen-Macaulay defect(both defined below).

This is described in 3.5, 3.6, and 3.7.

The definition of parameters for complexes is based on a notion ofanchor prime ideals. These do for complexes what minimal prime ideals do for mod- ules, and the quantitative relations between dimension and depth under dagger duality—studied in [3]—have a qualitative description in terms of anchor and associated prime ideals.

ThroughoutRdenotes a commutative, Noetherian local ring with maximal ideal_ᒊand residue fieldk=R/ᒊ. We use the same notation as in [4], but for convenience we recall a few basic facts.

The homological position and size of a complexXis captured by thesu-

Received May 2, 2000.

premum,infimum, andamplitude:

supX =sup{∈Z|H(X)=0},

infX =inf{∈Z|H(X)=0}, and ampX =supX−infX.

By convention, supX= −∞and infX= ∞if H(X)=0.

Thesupportof a complexXis the set

Supp_RX= {ᒍ∈SpecR|Xᒍ 0} =

Supp_RH(X).

As usual Min_RXis the subset of minimal elements in the support.

Thedepthand the (Krull) dimensionof an R-complex X are defined as follows:

depth_RX= −sup(RHom_R(k, X)), for X∈D−(R), and dim_RX=sup{dimR/ᒍ−infXᒍ|ᒍ∈Supp_RX},

cf. [6, Sec. 3]. For modules these notions agree with the usual ones. It follows from the definition that

(1.1) dim_RX≥dim_RᒍXᒍ+dimR/ᒍ

forX ∈D(R)andᒍ∈SpecR; and there are always inequalities:

−infX≤dim_RX for X∈D+(R); and (1.2)

−supX≤depth_RX for X∈D−(R).

(1.3)

A complex X ∈ Db^f(R) is Cohen-Macaulay if and only if dim_RX = depth_RX, that is, if an only if theCohen-Macaulay defect,

cmd_RX=dim_RX−depth_RX,

is zero. For complexes inDb^f(R)the Cohen-Macaulay defect is always non- negative, cf. [6, Cor. 3.9].

2. Anchor Prime Ideals

In [4] we introduced associated prime ideals for complexes. The analysis of the support of a complex is continued in this section, and the aim is now to identify the prime ideals that do for complexes what the minimal ones do for modules.

Definitions2.1. LetX ∈ D+(R); we say that_ᒍ ∈ SpecR is ananchor prime ideal forX if and only if dim_RᒍXᒍ = −infXᒍ > −∞. The set of anchor prime ideals forXis denoted by Anc_RX; that is,

Anc_RX= {ᒍ∈Supp_RX|dim_RᒍXᒍ+infXᒍ=0}.

Forn∈N0we set

Wn(X)= {ᒍ∈Supp_RX|dim_RX−dimR/ᒍ+infXᒍ≤n}.

Observation 2.2. Let S be a multiplicative system in R, and let ᒍ ∈ SpecR. If _ᒍ∩S = ∅ then S⁻¹ᒍ is a prime ideal in S⁻¹R, and for X ∈ D(R) there is an isomorphism S⁻¹XS⁻¹ᒍ Xᒍ in D(Rᒍ). In particular, infS⁻¹XS⁻¹ᒍ=infXᒍand dim_S⁻¹_RS−1ᒍS⁻¹XS⁻¹ᒍ=dim_RᒍXᒍ. Thus, the next biconditional holds forX ∈D+(R)andᒍ∈SpecRwithᒍ∩S= ∅.

(2.1) ᒍ∈Anc_RX ⇐⇒ S⁻¹ᒍ∈Anc_S⁻¹_RS⁻¹X.

Theorem2.3. ForX∈D+(R)there are inclusions:

Min_RX⊆Anc_RX; and (a)

W0(X)⊆Anc_RX.

(b)

Furthermore, if ampX = 0, that is, if X is equivalent to a module up to a shift, then

(c) Anc_RX=Min_RX⊆Ass_RX;

and ifX is Cohen-Macaulay, that is,X ∈ Db^f(R)and dim_RX = depth_RX, then

(d) Ass_RX⊆Anc_RX=W0(X).

Proof. In the followingXbelongs toD+(R).

(a): If ᒍ belongs to Min_RX then Supp_RᒍXᒍ = {ᒍ_ᒍ}, so dim_RᒍXᒍ =

−infXᒍ, that is,ᒍ_ᒍ∈Anc_RᒍXᒍand henceᒍ∈Anc_RXby (2.1).

(b): Assume that_ᒍbelongs to W0(X), then dim_RX = dimR/ᒍ−infXᒍ, and since dim_RX≥dim_RᒍXᒍ+dimR/ᒍand dim_RᒍXᒍ≥ −infXᒍ, cf. (1.1) and (1.2), it follows that dim_RᒍXᒍ= −infXᒍ, as desired.

(c): ForM ∈D0(R)we have

Anc_RM = {ᒍ∈Supp_RM |dim_RᒍMᒍ=0} =Min_RM,

and the inclusion Min_RM ⊆Ass_RMis well-known.

(d): Assume thatX ∈ Db^f(R)and dim_RX = depth_RX, then dim_RᒍXᒍ = depth_RᒍXᒍfor allᒍ∈Supp_RX, cf. [5, (16.17)]. Ifᒍ∈Ass_RXwe have

dim_RᒍXᒍ=depth_RᒍXᒍ= −supXᒍ≤ −infXᒍ,

cf. [4, Def. 2.3], and it follows by (1.2) that equality must hold, soᒍbelongs to Anc_RX.

For eachᒍ∈Supp_RXthere is an equality

dim_RX=dim_RᒍXᒍ+dimR/ᒍ,

cf. [5, (17.4)(b)], so dim_RX−dimR/ᒍ+infXᒍ=0 forᒍwith dim_RᒍXᒍ=

−infXᒍ. This proves the inclusion Anc_RX⊆W0(X). Corollary2.4.ForX ∈Db(R)there is an inclusion:

(a) Min_RX⊆Ass_RX∩Anc_RX;

and for_ᒍ∈Ass_RX∩Anc_RXthere is an equality:

(b) cmd_RᒍXᒍ=ampXᒍ.

Proof. Part (a) follows by 2.3 (a) and [4, Prop. 2.6]; part (b) is immediate by the definitions of associated and anchor prime ideals, cf. [4, Def. 2.3].

Corollary2.5.IfX∈D₊^f(R), then

dim_RX =sup{dimR/ᒍ+dim_RᒍXᒍ|ᒍ∈Anc_RX}.

Proof. It is immediate by the definitions that

dim_RX=sup{dimR/ᒍ−infXᒍ|ᒍ∈Supp_RX}

≥sup{dimR/ᒍ−infXᒍ|ᒍ∈Anc_RX}

=sup{dimR/ᒍ+dim_RᒍXᒍ|ᒍ∈Anc_RX};

and the opposite inequality follows by 2.3 (b).

Proposition2.6. The following hold:

(a) If X∈D+(R)and_ᒍbelongs toAnc_RX, thendim_Rᒍ(HinfXᒍ(Xᒍ))=0.

(b) If X∈Db^f(R), thenAnc_RXis a finite set.

Proof. (a): Assume thatᒍ∈Anc_RX; by [6, Prop. 3.5] we have

−infXᒍ=dim_RᒍXᒍ≥dim_Rᒍ(HinfXᒍ(Xᒍ))−infXᒍ,

and hence dim_Rᒍ(HinfXᒍ(Xᒍ))=0.

(b): By (a) every anchor prime ideal forXis minimal for one of the homo- logy modules ofX, and whenX ∈Db^f(R)each of the finitely many homology modules has a finite number of minimal prime ideals.

Observation2.7. By Nakayama’s lemma it follows that inf K(x¹, . . . , xn;Y )=infY, forY ∈D₊^f(R)and elementsx1, . . . , xn∈ᒊ.

Proposition2.8 (Dimension of Koszul Complexes). The following hold for a complexY ∈D₊^f(R)and elementsx1, . . . , xn∈ᒊ:

dim_RK(x¹, . . . , xn;Y ) (a)

=sup{dimR/ᒍ−infYᒍ|ᒍ∈Supp_RY ∩V(x¹, . . . , xn)}; and dim_RY −n≤dim_RK(x¹, . . . , xn;Y )≤dim_RY.

(b)

Furthermore:

The elementsx1, . . . , xnare contained in a prime ideal (c)

ᒍ∈W_n(Y ); and

dim_RK(x1, . . . , xn;Y )=dim_RY if and only ifx1, . . . , xn ∈ᒍ (d)

for some_ᒍ∈W0(Y ).

Proof. Since Supp_RK(x1, . . . , xn;Y ) = Supp_RY ∩V(x1, . . . , xn) (see [6, p. 157] and [4, 3.2]) (a) follows by the definition of Krull dimension and 2.7. In (b) the second inequality follows from (a); the first one is established through four steps:

1^◦ Y = R: The second equality below follows from the definition of Krull dimension as Supp_RK(x1, . . . , xn) = Supp_RH0(K(x1, . . . , xn)) = V(x1, . . . , xn), cf. [4, 3.2]; the inequality is a consequence of Krull’s Prin- cipal Ideal Theorem, see for example [8, Thm. 13.6].

dim_RK(x¹, . . . , xn;Y )=dim_RK(x¹, . . . , xn)

=sup{dimR/ᒍ|ᒍ∈V(x1, . . . , xn)}

=dimR/(x1, . . . , xn)

≥dimR−n

=dim_RY −n.

2^◦Y =B, a cyclic module: Byx¯1, . . . ,x¯nwe denote the residue classes in Bof the elementsx¹, . . . , xn; the inequality below is by 1^◦.

dim_RK(x1, . . . , xn;Y )=dim_RK(x¯1, . . . ,x¯n)

=dim_BK(x¯1, . . . ,x¯n)

≥dimB−n

=dim_RY−n.

3^◦ Y = H ∈ D0^f(R): We setB = R/Ann_RH; the first equality below follows by [6, Prop. 3.11] and the inequality by 2^◦.

dim_RK(x1, . . . , xn;Y )=dim_RK(x1, . . . , xn;B)

≥dim_RB−n

=dim_RY−n.

4^◦Y ∈Db^f(R): The first equality below follows by [6, Prop. 3.12] and the last by [6, Prop. 3.5]; the inequality is by 3^◦.

dim_RK(x1, . . . , xn;Y )=sup{dim_RK(x1, . . . , xn;H(Y ))−|∈Z}

≥sup{dim_RH(Y )−n−|∈Z}

=dim_RY −n.

This proves (b).

In view of (a) it now follows that

dim_RY−n≤dimR/ᒍ−infYᒍ

for someᒍ ∈ Supp_RY ∩V(x¹, . . . , xn). That is, the elementsx¹, . . . , xn are contained in a prime idealᒍ∈Supp_RY with

dim_RY −dimR/ᒍ+infYᒍ≤n, and this proves (c).

Finally, it is immediate by the definitions that

dim_RY =sup{dimR/ᒍ−infYᒍ|ᒍ∈Supp_RY∩V(x¹, . . . , xn)} if and only if W0(Y )∩V(x1, . . . , xn)= ∅. This proves (d).

Theorem2.9. IfY ∈Db^f(R), then the next two numbers are equal.

d(Y )=dim_RY +infY; and

s(Y )=inf{s ∈N₀| ∃x1, . . . , xs :_ᒊ∈Anc_RK(x1, . . . , xs;Y )}.

Proof. There are two inequalities to prove.

d(Y )≤s(Y ): Letx1, . . . , xs ∈ᒊbe such that_ᒊ∈Anc_RK(x1, . . . , xs;Y ); by 2.8 (b) and 2.7 we then have

dim_RY−s ≤dim_RK(x1, . . . , xs;Y )= −inf K(x1, . . . , xs;Y )= −infY, so d(Y )≤s, and the desired inequality follows.

s(Y ) ≤ d(Y ): We proceed by induction on d(Y). If d(Y ) = 0 thenᒊ ∈ Anc_RY so s(Y ) = 0. If d(Y ) > 0 thenᒊ ∈Anc_RY, and since Anc_RY is a finite set, by 2.6(b), we can choose an elementx ∈ᒊ− ∪ᒍ∈Anc_RYᒍ. We set K = K(x;Y ); it is cleat that s(Y ) ≤ s(K)+1. Furthermore, it follows by 2.8 (a) and 2.3 (b) that dim_RK <dim_RY and thereby d(K) <d(Y ), cf. 2.7.

Thus, by the induction hypothesis we have

s(Y )≤s(K)+1≤d(K)+1≤d(Y );

as desired.

3. Parameters

By 2.9 the next definitions extend the classical notions of systems and se- quences of parameters for finite modules (e.g., see [8, § 14] and the appendix in [2]).

Definitions3.1. LetY belong toDb^f(R)and setd = dim_RY +infY. A set of elementsx1, . . . , xd ∈ᒊare said to be asystem of parametersforY if and only ifᒊ∈Anc_RK(x1, . . . , xd;Y ).

A sequence x = x1, . . . , xn is said to be aY-parameter sequenceif and only if it is part of a system of parameters forY.

Lemma3.2. LetYbelong toDb^f(R)and setd =dim_RY+infY. The next two conditions are equivalent for elementsx¹, . . . , xd ∈ᒊ.

(i) x¹, . . . , xdis a system of parameters forY. (ii) For everyj ∈ {0, . . . , d}there is an equality:

dim_RK(x1, . . . , xj;Y )=dim_RY −j;

andxj+1, . . . , xdis a system of parameters forK(x1, . . . , xj;Y ).

Proof. (i)⇒(ii): Assume thatx1, . . . , xdis a system of parameters forY, then

−inf K(x1, . . . , xd;Y )=dim_RK(x1, . . . , xd;Y )

=dim_RK(xj+1, . . . , xd;K(x1, . . . , xj;Y )

≥dim_RK(x1, . . . , xj;Y )−(d−j) by 2.8 (b)

≥dim_RY −j−(d−j) by 2.8 (b)

=dim_RY −d

= −infY.

By 2.7 it now follows that−infY =dim_RK(x1, . . . , xj;Y )−(d−j), so dim_RK(x1, . . . , xj;Y )=d−j−infY =dim_RY −j, as desired. It also follows that d(K(x1, . . . , xj;Y ))=d−j, and since

ᒊ∈Anc_RK(x¹, . . . , xd;Y )=Anc_RK(xj+1, . . . , xd;K(x¹, . . . , xj;Y )), we conclude thatxj+1, . . . , xdis a system of parameters for K(x1, . . . , xj;Y ).

(ii)⇒(i): If dim_RK(x1, . . . , xj;Y )=dim_RY−jthen d(K(x1, . . . , xj;Y ))

= d−j; and ifxj+1, . . . , xdis a system of parameters for K(x1, . . . , xj;Y ) thenᒊbelongs to

Anc_RK(xj+1, . . . , xd;K(x1, . . . , xj;Y ))=Anc_RK(x1, . . . , xd;Y ), sox1, . . . , xdmust be a system of parameters forY.

Proposition3.3. LetY ∈Db^f(R). The following conditions are equivalent for a sequencex=x1, . . . , xnin_ᒊ.

(i) xis aY-parameter sequence.

(ii) For eachj ∈ {0, . . . , n}there is an equality:

dim_RK(x1, . . . , xj;Y )=dim_RY −j;

andxj+1, . . . , xnis aK(x1, . . . , xj;Y )-parameter sequence.

(iii) There is an equality:

dim_RK(x¹, . . . , xn;Y )=dim_RY −n.

Proof. It follows by 3.2 that (i) implies (ii), and (iii) follows from (ii). Now, setK = K(x;Y )and assume that dim_RK = dim_RY −n. Choose, by 2.9,

s =s(K)= dim_RK+infKelementsw1, . . . , ws in_ᒊsuch that_ᒊbelongs to Anc_RK(w¹, . . . , ws;K) = Anc_RK(x¹, . . . , xn, w¹, . . . , ws;Y ). Then, by 2.7, we have

n+s =(dim_RY −dim_RK)+(dim_RK+infK)=dim_RY +infY =d, sox¹, . . . , xn, w¹, . . . , ws is a system of parameters forY, whencex¹, . . . , xn

is aY-parameter sequence.

We now recover a classical result (e.g., see [2, Prop. A.4]):

Corollary 3.4. Let M be an R-module. The following conditions are equivalent for a sequencex=x1, . . . , xnin_ᒊ.

(i) xis anM-parameter sequence.

(ii) For eachj ∈ {0, . . . , n}there is an equality:

dim_RM/(x1, . . . , xj)M =dim_RM −j;

andxj+1, . . . , xnis anM/(x1, . . . , xj)M-parameter sequence.

(iii) There is an equality:

dim_RM/(x1, . . . , xn)M =dim_RM−n.

Proof. By [6, Prop. 3.12] and [5, (16.22)] we have dim_RK(x¹, . . . , xj;M)

=sup{dim_R(M⊗^L_RH(K(x1, . . . , xj)))−|∈Z}

=sup{dim_R(M⊗RH(K(x¹, . . . , xj)))−|∈Z}

=dim_R(M⊗RR/(x1, . . . , xj)).

Theorem3.5. Let Y ∈ Db^f(R). The following hold for a sequencex = x1, . . . , xnin_ᒊ.

(a) There is an inequality:

amp K(x;Y )≥ampY; and equality holds if and only ifxis aY-sequence.

(b) There is an inequality:

cmd_RK(x;Y )≥cmd_RY;

and equality holds if and only ifxis aY-parameter sequence.

(c) Ifxis a maximalY-sequence, then

ampY ≤cmd_RK(x;Y ).

(d) Ifxis a system of parameters forY, then cmd_RY ≤amp K(x;Y ).

Proof. In the followingKdenotes the Koszul complex K(x;Y ). (a): Immediate by 2.7 and [4, Prop. 5.1].

(b): By [4, Thm. 4.7 (a)] and 2.8 (b) we have

cmd_RK=dim_RK−depth_RK=dim_RK+n−depth_RY ≥cmd_RY, and by 3.3 equality holds if and only ifxis aY-parameter sequence.

(c): Supposexis a maximalY-sequence, then ampY =supY−infK by 2.7

= −depth_RK−infK by [4, Thm. 5.4]

≤cmd_RK by (1.2).

(d): Supposexis system of parameters forY, then ampK=supK+dim_RK

≥dim_RK−depth_RK by (1.3)

=cmd_RY by (b).

Theorem3.6. The following hold forY ∈Db^f(R). (a) The next four conditions are equivalent.

(i) There is a maximal Y-sequence which is also a Y-parameter se- quence.

(ii) depth_RY +supY ≤dim_RY+infY. (ii’) ampY ≤cmd_RY.

(iii) There is a maximal strongY-sequence which is also aY-parameter sequence.

(b) The next four conditions are equivalent.

(i) There is a system of parameters forY which is also aY-sequence.

(ii) dim_RY +infY ≤depth_RY+supY. (ii’) cmd_RY ≤ampY.

(iii) There is a system of parameters for Y which is also a strong Y- sequence.

(c) The next four conditions are equivalent.

(i) There is a system of parameters forY which is also a maximal Y- sequence.

(ii) dim_RY +infY =depth_RY +supY. (ii’) cmd_RY =ampY.

(iii) There is a system of parameters forYwhich is also a maximal strong Y-sequence.

Proof. Let Y ∈ Db^f(R), set n(Y ) = depth_RY + supY and d(Y ) = dim_RY +infY.

(a): A maximal Y-sequence is of length n(Y ), cf. [4, Cor. 5.5], and the length of aY-parameter sequence is at most d(Y ). Thus, (i) implies (ii) which in turn is equivalent to (ii’). Furthermore, a maximal strongY-sequence is, in particular, a maximalY-sequence, cf. [4, Cor. 5.7], so (iii) is stronger than (i). It is now sufficient to prove the implication (ii)⇒(iii): We proceed by induction.

If n(Y ) = 0 then the empty sequence is a maximal strongY-sequence and a Y-parameter sequence. Let n(Y ) > 0; the two sets Ass_RY and W0(Y ) are both finite, and since 0 < n(Y ) ≤ d(Y ) none of them containᒊ. We can, therefore, choose an elementx ∈ᒊ− ∪Ass_RY∪W0(Y )ᒍ, andxis then a strong Y-sequence, cf. [4, Def. 3.3], and aY-parameter sequence, cf. 3.3 and 2.8. Set K = K(x;Y ), by [4, Thm. 4.7 and Prop. 5.1], respectively, 2.8 and 2.7 we have

depth_RK+supK=n(Y )−1≤d(Y )−1=dim_RK+infK.

By the induction hypothesis there exists a maximal strongK-sequencew¹, . . . , wn−1which is also aK-parameter sequence, and it follows by [4, 3.5] and 3.3 thatx, w1, . . . , wn−1is a strongY-sequence and aY-parameter sequence, as wanted.

The proof of (b) i similar to the proof of (a), and (c) follows immediately by (a) and (b).

Theorem3.7. The following hold forY ∈Db^f(R):

(a) If ampY =0, then anyY-sequence is aY-parameter sequence.

(b) If cmd_RY =0, then anyY-parameter sequence is a strongY-sequence.

Proof. The empty sequence is aY-parameter sequence as well as a strong Y-sequence, this founds the base for a proof by induction on the lengthnof

the sequencex=x1, . . . , xn. Letn >0 and setK=K(x1, . . . , xn−1;Y ); by 2.8 (a) we have

(∗)

dim_RK(x1, . . . , xn;Y )=dim_RK(xn;K)

=sup{dimR/ᒍ−infKᒍ|ᒍ∈Supp_RK∩V(xn)}.

Assume that ampY = 0. Ifxis aY-sequence, then ampK =0 by 3.5 (a) andxn ∈z_RK, cf. [4, Def. 3.3]. As z_RK= ∪ᒍ∈AssRKᒍ, cf. [4, 2.5], it follows by (b) and (c) in 2.3 thatxnis not contained in any prime idealᒍ∈W0(K); so from(∗)we conclude that dim_RK(xn;K) <dim_RK, and it follows by 2.8 (b) that dim_RK(xn, K) = dim_RK−1. By the induction hypothesis dim_RK = dim_RY−(n−1), so dimRK(x1, . . . , xn;Y )=dim_RY −nand it follows by 3.3 thatxis aY-parameter sequence. This proves (a).

We now assume that cmd_RY =0. Ifxis aY-parameter sequence then, by the induction hypothesis,x1, . . . , xn−1is a strongY-sequence, so it is sufficient to prove thatxn∈Z_RK, cf. [4, 3.5]. By 3.3 it follows thatxnis aK-parameter sequence, so dim_RK(xn;K) = dim_RK−1 and we conclude from(∗)that xn ∈ ∪ᒍ∈W0(K)ᒍ. Now, by 3.5 (b) we have cmd_RK = 0, so it follows from 2.3 (d) thatxn ∈ ∪ᒍ∈Ass_RKᒍ=Z_RK. This proves (b).

Semi-dualizing Complexes3.8. We recall two basic definitions from [3]:

A complexC ∈Db^f(R)is said to besemi-dualizingforRif and only if the homothety morphismχ_C^R:R →RHom_R(C, C)is an isomorphism [3, (2.1)].

LetC be a semi-dualizing complex forR. A complexY ∈ Db^f(R)is said to beC-reflexiveif and only if thedagger dualY^†C =RHom_R(Y, C)belongs toDb^f(R)and the biduality morphismδ^C_Y:Y →RHom_R(RHom_R(Y, C), C) is invertible inD(R)[3, (2.7)].

Relations between dimension and depth forC-reflexive complexes are stud- ied in [3, sec. 3], and the next result is an immediate consequence of [3, (3.1) and (2.10)].

Let C be a semi-dualizing complex for R and let Z be aC-reflexive complex. The following holds for ᒍ ∈ SpecR: If ᒍ ∈ Anc_RZ then ᒍ∈Ass_RZ^†C, and the converse holds inCis Cohen-Macaulay.

Adualizing complex, cf. [7], is a semi-dualizing complex of finite injective dimension, in particular, it is Cohen-Macaulay, cf. [3, (3.5)]. IfDis a dualizing complex forR, then, by [7, Prop. V.2.1], all complexesY ∈ Db^f(R)areD- reflexive; in particular, all finiteR-modules areD-reflexive and, therefore, [4, 5.10] is a special case of the following:

Theorem3.9. LetCbe a Cohen-Macaulay semi-dualizing complex forR, and letx = x¹, . . . , xn be a sequence in_ᒊ. IfY isC-reflexive, thenx is a Y-parameter sequence if and only if it is aRHom_R(Y, C)-sequence; that is

xis aY-parameter sequence ⇐⇒ xis aRHom_R(Y, C)-sequence.

Proof. We assume thatC is a Cohen-Macaulay semi-dualizing complex forRand thatY isC-reflexive, cf. 3.8. The desired biconditional follows by the next chain, and each step is explained below (we use the notation −^†C introduced in 3.8).

xis aY-parameter sequence ⇐⇒ cmd_RK(x;Y )=cmd_RY

⇐⇒ amp K(x;Y )^†C =ampY^†C

⇐⇒ amp K(x;Y^†C)=ampY^†C

⇐⇒ xis aY^†C-sequence.

The first biconditional follows by 3.5 (b) and the last by 3.5 (a). Since K(x)is a bounded complex of free modules (hence of finite projective dimension), it follows from [3, Thm. (3.17)] that also K(x;Y )isC-reflexive, and the second biconditional is then immediate by the CMD-formula [3, Cor. (3.8)]. The third one is established as follows:

K(x;Y )^†C RHom_R(K(x)⊗^L_RY, C) RHom_R(K(x), Y^†C) RHom_R(K(x), R⊗^L_RY^†C) RHom_R(K(x), R)⊗^L_RY^†C

∼K(x)⊗^L_RY^†C K(x;Y^†C),

where the second isomorphism is by adjointness and the fourth by, so- called, tensor-evaluation, cf. [1, (1.4.2)]. It is straightforward to check that Hom_R(K(x), R)is isomorphic to the Koszul complex K(x)shiftedndegrees to the right, and the symbol∼denotes isomorphism up to shift.

IfCis a semi-dualizing complex forR, then bothCandRareC-reflexive complexes, cf. [3, (2.8)], so we have an immediate corollary to the theorem:

Corollary3.10. IfC is a Cohen-Macaulay semi-dualizing complex for R, then the following hold for a sequencex=x1, . . . , xnin_ᒊ.

(a) xis aC-parameter sequence if and only if it is anR-sequence.

(b) xis anR-parameter sequence if and only if it is aC-sequence.

Acknowledgements.The author would like to thank professor Srikanth Iyengar and professor Hans-Bjørn Foxby for taking the time to discuss the material presented here.

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